The notation $A^k$ means a repeated matrix product between $k$ copies of the square matrix $A$.

1. Assume $A$ is an $n\times n$ matrix where $A^2=\zeromatrix$ (which does not imply that $A=\zeromatrix$.) Prove that $I_n-A$ is invertible by showing that $I_n+A$ is an inverse of $I_n-A$.
2. Assume that $A$ is an $n\times n$ matrix where $A^3=\zeromatrix$. Prove that $I_n-A$ is invertible.
3. Form a general theorem based on your observations from parts (1) and (2) and provide a proof.